Now, although the predictions provided by the model are in good agreement with the experimental results,
obtaining such predictions is not a simple task since the field equations that emerge from the NLSM are in general very complicated. In fact, these are a set of \((N^2-1)\)  coupled non-linear differential equations
-being \(N\)  the flavor number encoded in the \(SU(N)\)  group- which explains that most of the known solutions have been constructed numerically. Some relevant results in this approach are found in Refs. [1]}, [2]}, [3]} and [4]}. Of course, when the theory is coupled to general relativityWhen a suitable interaction potential is considered some interesting analytic solutions can be constructed; see [5]}. or the Maxwell theory to describe more complex physical objects and processes, the field equations become even more intrincate. However, in Refs. [6]} and [7]}, suitable ansätze have been introduced that go beyond
spherical symmetry (collectively called generalized hedgehog ansatz),
allowing to find a good number of exact solutions not only in the NLSM, but also in the Skyrme model,
the generalized Skyrme model and the Yang-Mills-Higgs theory.
These new set of solutions describe boson stars [8]}, black holes [7]}, [10]}, [11]},See [12]} for the analysis of the lensing produced by these Skyrme black holes. black strings [13]}, gravitating solitons
[14]}, [15]}, topological solitons at finite volume [16]}, and even crystalline structures of topological solitons [17]}, [18]} (see also [19]}, [20]}, [21]}, [22]}, [23]}, [24]} and references therein).
